Problem

$f(x) = x^2 - \frac{7x - 4\sqrt{x}}{x^3}\quad x > 0$

(a) Show that the equation $f(x) = 0$ has a root $\alpha$ in the interval $[0.3, 0.4]$

(b) Determine $f'(x)$.

(c) Using $x_0 = 0.3$ as a first approximation for $\alpha$, apply the Newton–Raphson procedure once to $f(x)$ to determine a second approximation for $\alpha$, giving your answer to 3 decimal places.

The equation $f(x) = 0$ has another root $\beta$ in the interval $[1.3, 1.5]$

(d) Use linear interpolation once on the interval $[1.3, 1.5]$ to determine an approximation for $\beta$, giving your answer to 3 decimal places.